Ch 3.2A Notes Name: Least-Squares Regression Key

Statistics ? Ch 3.2A Notes

Key Name: ___________________________

Least-Squares Regression

Regression Line Everyone knows that cars and trucks lose value the more they are driven. Can we predict the price of a used Ford F-150 SuperCrew 4X4 if we know how many miles it has on the odometer? A random sample of 16 used Ford F-150 SuperCrew 4X4s was selected from among those listed for sale at . The number of miles driven and price (in dollars) were recorded for each of the trucks. Here are the data.

Miles 70,583 129,484 29,932 29,953 24,495 75,678 8,359 4,447 34,077 58,023 44,447 68,474 144,162 140,776 29,397 131,385

Price 21,994 9,500 29,875 41,995 41,995 28,986 31,891 37,991 34,995 29,988 22,896 33,961 16,883 20,897 27,495 13,997

Linear (straight-line) relationships between two quantitative variables are common and easy to

understand. A regression line summarizes the __r_e__la__t_i_o__n_s__h__i_p_______ between two variables, but

only in settings where one of the variables helps explain or predict the other.

Line Aex_p_Rla_en_a_gt_or_re_y_sv_sa_ri_ioa_bn_l_e_x_c_h_a_n_g_e_s_._W__e_o_f_teisnauslienea

that describes how a response variable y changes as an regression line to predict the value of y for a given value

of x.

model A regression line is a _________________ for the data, much like density curves. The equation of a

regression line gives a compact mathematical description of what this model tells us about the relationship between the response variable y and the explanatory variable x.

Suppose that y is a response variable and x is an explanatory variable. A regression line relating y to x has an equation of the form

= a + bx In this equation,

value ? (read "y hat") is the _p__r_e__d__i_c__t_e_d__________of the response variable y for a given value

of the explanatory variable x.

slope ? b is the _________________, the amount by which y is predicted to change when x increases by one unit.

g intercept ? a is the __________________, the predicted value of y when x = 0.

Statistics ? Ch 3.2A Notes

Name: ___________________________ Least-Squares Regression

Example Problem: Interpreting slope and y-intercepts The equation of the regression line is shown

price = 38257 0.1629(miles driven)

1. Identify the slope and y-intercept of the regression line.

Slope o b o 1629

Y int o a 38,257

2. Interpret each value in context.

fpaborebrTedeTihacehctenedpehdrratpeidovdrdgeiicicntotiezdoeendooprawolrfmnimcbaieilulyeoestfsoeha1defi6ftsoor2ur93rdcdd8kof,hfl2lIaas51rso5s7b0te1Nh6eiasen4tdwhrtaivrsuenck

3. Use the regression line to predict the price for a Ford F-150 with 100,000 miles driven.

price 38257 o1629 tooooo price 21967

__E__x_t_r__a_p__o__la__t_i_o_n___________ is the use of a regression line for prediction far outside the interval

of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate.

Don't make predictions using values of x that are much larger or much smaller than those that actually appear in your data.

Practice Problem: Some data were collected on the weight of a male white laboratory rat for the first 25 weeks after its birth. A scatterplot of the weight (in grams) and time since birth (in weeks) shows a fairly strong, positive linear relationship. The linear regression equation = 100 + 40() models the data fairly well.

1. What is the slope of the regression line? Explain what this means in context.

b 40j foreachadditional week we predictthat a rat willgain 40grams of weight

2. What is the y-intercept? Explain what this means in context

a loo Thepredictedweight of a new born rat is 100grams

3. Predict the rat's weight after 16 weeks. Show your work.

weight 100 t 4046

weight 740grams

Statistics ? Ch 3.2A Notes

Name: ___________________________ Least-Squares Regression

4. Should you use this line to predict the rat's weight at age 2 years? Use the equation to make

the prediction and think about the reasonableness of the result. (there are 454 grams in a

pound) 2years 104weeks

weight loot 40404

This is unreasonable and is

theresultof extrapolation

weight 4,260grams

or

Residuals

9 4lbs

In most cases, no line will pass exactly through all the points in a scatterplot. A good regression line

makes the vertical distances of the points from the line as small as possible.

residual A _________________ is the difference between an observed value of the response variable and the

value predicted by the regression line.

residual = _o_b_s__e_r_v_e__d_ ? _p_r_e__d_i_c_t_e__d

y

Other Names for Regression Line: x Least Squares Regression Line x LSRL x Linear Model x Regression Equation

Practice Problem: Find and interpret

Finding a Residual the residual for the

Ford

F-150

that

had

70X, 583

miles

driven

and

a

price

of

$21Y, 994.a

observer

D price 38257 0.162970.583 price 38257 0.1629(miles driven)

price 26,759 residuals observed Predicted predicted residual 21,994 26,759

residual 4,765

Theactualprice of this truck

was 4 65 lowerthanexpected pbarisceemdonigithstbmeilloewageeraTshaearecstuulat l ofmanyotherfactors

Least Squares Regression Line

Different regression lines produce different residuals. The regression line we want is the one that

minimizes the sum of the squared residuals.

squared The least-squares regression line of y on x is the line that makes the sum of the ________________

residuals ____________________ as small as possible.

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